http://en.wikipedia.org/wiki/Dixieland#Chicago_style
It really makes a difference when & who you are talking about etc... For example, wynton marsalis currently plays in chicago & new york but is from new orleans.
Here is his father Ellis marsalis, recorded in new orleans in 1963 with mostly new orleans crew: http://www.amazon.com/gp/product/B000QQWHAY/ To me, it sounds more like it's from a particular time than a particular place, although James Black's rhythm could surely be singled out as "new orleans" in some places.
I think shortly after this recording the group left the city & went their separate ways because (mainly i think) it was very hard to make a living in the segregated south as a black musician.... This was probably just one of many reasons to leave a segregated south & head to NY or Chicago where there were other communities. So if jazz started in new orleans, the city missed an opportunity to keep the art form growing & evolving there when it was unable to give the artists the respect/wages they deserved when they realized their worth. The situation is probably different there now, but I think this unfortunate circumstance has significantly shaped regional styles. The difference in styles over time between the cities you've mentioned reflects where artists were while innovating - pushing the boundary to the next natural step in the music. Of course once a community of a certain style is founded somewhere, people who want to play that type of music will probably go there & further increase the size of the community.